Andreas Klose

Algorithms for solving the single-sink fixed-charge transportation problem

The single-sink fixed-charge transportation problem is an important subproblem of the fixed-charge transportation problem. Just a few methods have been proposed in the literature to solve this problem. In this paper, solution approaches based on dynamic programming and implicit enumeration are revisited. It is shown how the problem size as well as the search space of a recently published dynamic programming method can be reduced by exploiting reduced cost information. Additionally, a further implicit enumeration approach relying on solution concepts for the binary knapsack problem is introduced. The performance of the various solution methods is compared in a series of computational experiments.

Demand dispersion and logistics costs in one-to-many distribution systems

When an organization decides on which groups of consumers it should target, the locations of these target consumers often play a role. Methods from the field of market segmentation are able to identify target groups with high benefit levels, but the expected costs of supplying products to the target groups are less well understood. These costs can play a large role if the locations of the customers, the demand locations, are geographically widely dispersed. This paper focuses on one-to-many distribution systems in which a central facility serves all demand points. We derive accurate logistics cost estimates from the dispersion of demand points for such systems, enabling a comparison of the expected logistics costs of different candidate target groups. The most accurate measure combines the average distance from the demand locations to a central location and the mutual distances between neighboring demand locations. The average of the distances between all pairs of locations forms a good alternative measure.

Analysis of some greedy algorithms for the single-sink fixed-charge transportation problem

Abstract The single-sink fixed-charge transportation problem (SSFCTP) consists of finding a minimum cost flow from a number of nodes to a single sink. Beside a cost proportional to the amount shipped, the flow cost encompass a fixed charge. The SSFCTP is an important subproblem of the well-known fixed-charge transportation problem. Nevertheless, just a few methods for solving this problem have been proposed in the literature. In this paper, some greedy heuristic solutions methods for the SSFCTP are investigated. It is shown that two greedy approaches for the SSFCTP known from the literature can be arbitrarily bad, whereas an approximation algorithm proposed in the literature for the binary min-knapsack problem has a guaranteed worst case bound if adapted accordingly to the case of the SSFCTP.

A Simple but Usually Fast Branch-and-Bound Algorithm for the Capacitated Facility Location Problem

This paper presents a simple branch-and-bound method based on Lagrangean relaxation and subgradient optimization for solving large instances of the capacitated facility location problem (CFLP) to optimality. To guess a primal solution to the Lagrangean dual, we average solutions to the Lagrangean subproblem. Branching decisions are then based on this estimated (fractional) primal solution. Extensive numerical results reveal that the method is much faster and more robust than other state-of-the-art methods for solving the CFLP exactly.

On line Routing per Mobile Phone A Case on Subsequent Deliveries of Newspapers

On-line routing is concerned with building vehicle routes in an on-going fashion in such a way that customer requests arriving dynamically in time are efficiently and effectively served. An indispensable prerequisite for applying on-line routing methods is mobile communication technology. Additionally it is of utmost importance that the employed communication system is suitable integrated with the firm’s enterprise application system and business processes. On basis of a case study, we describe in this paper a system that is cheap and easy to implement due to the use of simple mobile phones. Additionally, we address the question how on-line routing methods can be integrated in this system.

Solving the Single-Sink, Fixed-Charge, Multiple-Choice Transportation Problem by Dynamic Programming

This paper considers a minimum-cost network flow problem in a bipartite graph with a single sink. The transportation costs exhibit a staircase cost structure because such types of transportation cost functions are often found in practice. We present a dynamic programming algorithm for solving this so-called single-sink, fixed-charge, multiple-choice transportation problem exactly. The method exploits heuristics and lower bounds to peg binary variables, improve bounds on flow variables, and reduce the state-space variable. In this way, the dynamic programming method is able to solve large instances with up to 10,000 nodes and 10 different transportation modes in a few seconds, much less time than required by a widely used mixed-integer programming solver and other methods proposed in the literature for this problem.

An improved Lagrangian relaxation and dual ascent approach to facility location problems

The literature knows semi-Lagrangian relaxation as a particular way of applying Lagrangian relaxation to certain linear mixed integer programs such that no duality gap results. The resulting Lagrangian subproblem usually can substantially be reduced in size. The method may thus be more efficient in finding an optimal solution to a mixed integer program than a “solver” applied to the initial MIP formulation, provided that “small” optimal multiplier values can be found in a few iterations. Recently, a simplification of the semi-Lagrangian relaxation scheme has been suggested in the literature. This “simplified” approach is actually to apply ordinary Lagrangian relaxation to a reformulated problem and still does not show a duality gap, but the Lagrangian dual reduces to a one-dimensional optimization problem. The expense of this simplification is, however, that the Lagrangian subproblem usually can not be reduced to the same extent as in the case of ordinary semi-Lagrangian relaxation. Hence, an effective method for optimizing the Lagrangian dual function is of utmost importance for obtaining a computational advantage from the simplified Lagrangian dual function. In this paper, we suggest a new dual ascent method for optimizing both the semi-Lagrangian dual function as well as its simplified form for the case of a generic discrete facility location problem and apply the method to the uncapacitated facility location problem. Our computational results show that the method generally only requires a very few iterations for computing optimal multipliers. Moreover, we give an interesting economic interpretation of the semi-Lagrangian multiplier(s).

A bi-objective approach to discrete cost-bottleneck location problems

This paper considers a family of bi-objective discrete facility location problems with a cost objective and a bottleneck objective. A special case is, for instance, a bi-objective version of the (vertex) p-centdian problem. We show that bi-objective facility location problems of this type can be solved efficiently by means of an

The Single-Sink Fixed-Charge Transportation Problem: Applications and Solution Methods

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